Artificial boundary conditions for random ellitpic systems with correlated coefficient field

TL;DR

This paper develops and analyzes numerical algorithms for computing electrical fields in infinite, correlated random media, demonstrating their effectiveness and convergence properties through theoretical estimates and simulations.

Contribution

It extends boundary condition techniques to correlated media, providing new semi-group estimates and constructing sub-linear second-order correctors.

Findings

Algorithm performs well in correlated media with high probability.

Convergence rates depend on correlation size and domain parameters.

Numerical simulations support theoretical optimality.

Abstract

We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale $l$ in an infinite heterogeneous correlated random medium, in a situation where the medium is only known in a box of diameter $L\gg l$ around the support of the charge. We show that the algorithm of Lu, Otto and Wang, suggesting optimal Dirichlet boundary conditions motivated by the multipole expansion of Bella, Giunti and Otto, still performs well in correlated media. With overwhelming probability, we obtain a convergence rate in terms of $l$, $L$ and the size of the correlations for which optimality is supported with numerical simulations. These estimates are provided for ensembles which satisfy a multi-scale logarithmic Sobolev inequality, where our main tool is an extension of the semi-group estimates established by the first author. As part of our strategy, we construct sub-linear second-order correctors in this correlated setting which is of independent interest.